Mathematics for Physics. 2

Stone M.

Table of contents :
Mathematics for Physics II……Page 1
Preface……Page 3
Contents……Page 5
1.1 Covariant and Contravariant Vectors……Page 9
1.2.1 Transformation Rules……Page 12
Tensor algebra……Page 13
1.2.2 Tensor Product Spaces……Page 14
Tensor Products and Quantum Mechanics……Page 16
1.2.3 Symmetric and Skew-symmetric Tensors……Page 17
Bosons and Fermions……Page 19
1.2.4 Tensor Character of Linear Maps and Quadratic Forms……Page 20
1.2.5 Numerically Invariant Tensors……Page 22
1.3.1 Stress and Strain……Page 24
1.3.2 The Maxwell Stress Tensor……Page 30
2.1 Vector Fields and Covector Fields……Page 33
2.2.1 Lie Bracket……Page 38
Frobenius’ Theorem……Page 40
2.2.2 Lie Derivative……Page 41
2.3.1 Differential Forms……Page 44
2.3.2 The Exterior Derivative……Page 45
Cartan’s formulae……Page 47
Lie Derivative of Forms……Page 48
2.4.1 Maxwell’s Equations……Page 49
2.4.2 Hamilton’s Equations……Page 53
The classical mechanics of spin……Page 56
2.5.1 Connections……Page 58
2.5.2 Cartan’s Viewpoint: Local Frames……Page 59
3.1.1 Line Integrals……Page 61
3.1.2 Skew-symmetry and Orientations……Page 62
Orientable Manifolds……Page 63
3.2.1 Counting Boxes……Page 64
3.2.2 General Case……Page 65
The volume form……Page 67
3.3 Stokes’ Theorem……Page 68
3.4.1 Pull-backs and Push-forwards……Page 70
3.4.2 Spin textures……Page 72
3.4.3 The Hopf Map……Page 74
3.4.4 The Hopf Linking Number……Page 77
4.1 A Topological Miscellany……Page 83
Homeomorphism and Diffeomorphism……Page 84
4.2.1 Retractable Spaces: Converse of Poincare Lemma……Page 85
4.2.2 De Rham Cohomology……Page 88
4.3.1 Chains, Cycles and Boundaries……Page 89
Simplicial Complexes……Page 90
Chains……Page 92
The Boundary Operator……Page 93
Cycles, Boundaries and Homology……Page 94
The Euler Character……Page 97
4.3.2 De Rham’s Theorem……Page 99
4.4 Hodge Theory and the Morse Index……Page 104
4.4.1 The Laplacian on p-forms……Page 105
4.4.2 Morse Theory……Page 109
Supersymmetric Quantum Mechanics……Page 110
The Weitzenb.ock Formula……Page 115
5.1.1 Group Axioms……Page 119
Examples of Groups:……Page 120
5.1.2 Elementary Properties……Page 121
5.1.3 Group Actions on Sets……Page 125
5.2 Representations……Page 126
Direct Sum and Direct Product……Page 128
Schur’s Lemma……Page 129
Unitary Representations of Finite Groups……Page 130
Orthogonality of the Matrix Elements……Page 131
Class Functions and Characters……Page 132
5.2.3 The Group Algebra……Page 133
Projection Operators……Page 134
Real and Pseudoreal representations……Page 127
5.3.1 Vibrational spectrum of H2O……Page 136
5.3.2 Crystal Field Splittings……Page 140
6.1 Matrix Groups……Page 143
The Unitary group……Page 144
6.1.2 Symplectic Groups……Page 145
Unitary Symplectic Group……Page 147
6.2 Geometry of SU(2)……Page 148
6.2.1 Invariant vector elds……Page 150
Right-invariant vector elds……Page 151
6.2.2 Maurer-Cartan Forms……Page 152
6.2.3 Euler Angles……Page 154
6.2.4 Volume and Metric……Page 155
6.2.5 SO(3) approx SU(2)/Z2……Page 156
The Adjoint Representation……Page 160
6.2.6 Peter-Weyl Theorem……Page 161
6.2.7 Lie Brackets vs. Commutators……Page 163
6.3 Abstract Lie Algebras……Page 164
6.3.2 The Killing form……Page 166
6.3.3 Roots and Weights……Page 167
SU(2)……Page 168
SU(3)……Page 169
6.3.4 Product Representations……Page 175
Ideals and Quotient algebras……Page 165
7.1 Cauchy-Riemann equations……Page 177
7.1.1 Conjugate pairs……Page 179
7.1.2 Conformal Mapping……Page 183
The Riemann Mapping Theorem……Page 185
7.2.1 The Complex Integral……Page 187
7.2.2 Cauchy’s theorem……Page 189
7.2.3 The residue theorem……Page 192
7.3.1 Two-dimensional vector calculus……Page 195
7.3.2 Milne-Thomson Circle Theorem……Page 197
7.3.3 Blasius and Kutta-Joukowski Theorems……Page 198
7.4.1 Cauchy’s Integral Formula……Page 202
Liouville’s Theorem……Page 203
7.4.2 Taylor and Laurent Series……Page 204
Morera’s Theorem……Page 206
Taylor’s Theorem……Page 207
Laurent Series……Page 208
7.4.3 Zeros and Singularities……Page 209
7.4.4 Analytic Continuation……Page 210
The distribution x_+^alfa-1……Page 211
Weierstrass-Casorati……Page 214
7.5 Meromorphic functions and the Winding-Number……Page 215
Local mapping theorem……Page 216
7.5.2 Rouche’s theorem……Page 217
7.6.1 The Point at Innity……Page 219
7.6.2 Logarithms and Branch Cuts……Page 222
7.6.3 Conformal Coordinates……Page 229
The Jacobean torus……Page 231
Rational Trigonometric Expressions……Page 233
Rational Functions……Page 234
8.1.2 Branch-cut integrals……Page 235
8.1.3 Jordan’s Lemma……Page 238
8.2 The Schwarz Re ection Principle……Page 244
8.2.1 Kramers-Kronig Relations……Page 248
8.2.2 Hilbert transforms……Page 251
8.3.1 Mittag-Leer Partial-Fraction Expansion……Page 253
8.3.2 Infinite Product Expansions……Page 255
Convergence of Infinite Products……Page 256
8.4.1 Wiener-Hopf Sum Equations……Page 257
9.1 The Gamma Function……Page 263
Infinite Product for……Page 267
9.2.1 Monodromy……Page 268
9.2.2 Hypergeometric Functions……Page 269
9.3 Solving ODE’s via Contour integrals……Page 273
9.3.1 Bessel Functions……Page 276
9.4 Asymptotic Expansions……Page 279
9.4.1 Stirling’s Approximation for n!……Page 282
9.4.2 Airy Functions……Page 283
9.5 Elliptic Functions……Page 290
modular invariance……Page 294